7. Rafael Nunez’s article

Mathematical Idea Analysis: What embodied cognitive science can say about the human nature of mathematics---Rafael E. Nunez

Assignment requirement: Write a short (imaginary) letter from Nunez to a skeptical math teacher who believes that math has nothing to do with the body or the physical world. In the letter, Nunez should be trying to convince the skeptical teacher of his ideas.

Dear Mr. Wang:

I still remember the issues about mathematics education that we discussed last time when we first met each other. Your comments on what the nature of mathematics is got me think further. From discussing with you, I know that you, as a secondary school math teacher, have been suspecting for quite a long time about a problem that is whether or not math has anything to do with the body or the physical world. In your view, not all of the math concepts / ideas can be explained and understood explicitly, for instance, you illustrated an example “Why the empty set is a subset of all sets?”, the question posed by your students. Your reply was that “The empty set is a subset of all sets” is merely a definition and asked your students to memorize it rather than understand its true meaning. As to this point, I totally disagree with you. So, this is the main reason why I’m writing to you now.

From my perspective, mathematics is neither transcendentally objective nor arbitrary. The nature of mathematics is about human ideas, not just about formal proofs, axioms, and definitions. These human ideas are grounded in species-specific everyday cognitive and bodily mechanisms. In the last century, many influential mathematicians viewed human intuitions (which are not theorems, axioms, and definitions) as basis for helping explain and understand mathematics. These intuitions, although they were not proved scientifically, demonstrated implicitly that mathematics is based on aspects of the human mind. In recent years, there are some empirical findings about the nature of mind discovered based on the contemporary embodied cognitive science. These empirical findings show that we are able to understand the real meanings of mathematical ideas through bodily mechanisms or people’s physical world.

In this letter, I will draw on Mathematical Idea Analysis which comes out of the embodied cognitive science as a theoretical tool to further address why we can understand mathematical ideas through bodily, physical mechanisms and people’s everyday experience embodied by image schemas, conceptual metaphors (this one is very important, constituting the very fabric of mathematics), and etc in the following:

First, by using Mathematical Idea Analysis, I have found that mathematical ideas are grounded in bodily-based mechanisms and everyday experience. For example, we can use the everyday concept of a collection of objects in a bounded region of space to conceptualize the math concept of a set; use the everyday concept of a repeated action to conceptualize the math concept of recursion; use the everyday concept of rotation to conceptualize the math concept of complex arithmetic; the everyday concepts as motion, approaching a boundary, etc to conceptualize derivatives in calculus…

Second, conceptual metaphors are fundamental cognitive mechanisms which project the inferential structure of a source domain onto a target domain, allowing the use of effortless species-specific body-based inference to structure abstract inference. They are not arbitrary, because they are structured by species-specific constrains underlying our everyday experience- especially bodily experience. For example, Affection Is Warmth, here, Affection is conceptualized in terms of thermic experience. The same happens in mathematics. For instance, Classes (Sets) Are Container Schemas—Grounding metaphors; use ‘more than’ and ‘as many as’ for infinite sets—Redefinitional metaphors; Functions Are Sets of Points, the Sets Are Graphs—Linking metaphors (see the details of three types of conceptual metaphors in Lakoff & Nunez (2000)).

Third, image schemas are basic dynamic topological and orientation structures that characterize spatial inferences and link language to visual-motor experience. Their inferential structure is preserved under metaphorical mappings. So, if we go back to the instance ‘Sets Are Container Schemas’, we can see that the Container Schema(a common image schema in mathematics) presents the link between language and spatial perception, involving three parts: an Interior, a Boundary, and an Exterior. Image schemas can fit visual perception, for example, Venn Diagrams are visual instantiations of Container Schemas.

Of course, conceptual metaphors are closely related to image schemas. Now, please allow me to still focus on the example of “Sets Are Container Schemas”, here, sets initially metaphorically conceptualized as containers, and then Venn Diagrams work as symbolizations of sets to show their bounded regions in space. Therefore, for this mathematical idea ‘Sets’, we can realize the understanding through metaphorically relating them to containers and then using Venn Diagram as the image schema for containment. It means that we can conceptualize ‘Sets’ through our everyday life experience and body-based perception.

Furthermore, the ‘Laws of Container Schemas’ called by Lakoff and I are conceptual in nature and are reflections at the cognitive level based on the analysis of neural structures. So, I sincerely invite you to read my paper which includes more regarding this issue.

Yours sincerely,